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Talk:Zermelo–Fraenkel set theory
I suggest moving this to Zermelo–Fraenkel set theory and just add mention in the article that it's differentiated based on appearance of AC. Length of the title is quite ridiculous. :Done you're.so. 21:03, August 26, 2014 (UTC) ::Isn't there a mistake in the second axiom? It says if w is in some set z, that it is then in x or in y for every set x and for every set y. What if x and y are empty set? Shouldn't it say something about the relation between x,y,z? Wythagoras (talk) 17:39, August 27, 2014 (UTC) :::I think you are reading quantifiers backwards. Suppose we have two sets, x and y. Axiom says that then we have a set z which has as members x, y and nothing more. That is to say, if w is element of z, then it's either x or y. So z is desired pair {x,y}. LittlePeng9 (talk) 19:55, August 27, 2014 (UTC) Googological? The article doesn't contain any information of how it is related to googology and can produce numbers. It deserves the place on math wiki, but not here. Ikosarakt1 (talk ^ ) 05:51, August 31, 2014 (UTC) Yes,but set theory has been and can be used for creating very large numbers.It is used in the definition of Rayo's number,Fish number 7 and BIG FOOT.It is also important for large ordinals and cardinals,both used in hierarchies mesuring the strenght of fast-growing functions and big numbers.Boboris02 (talk) 20:27, November 12, 2016 (UTC) Strength Which system is the strongest among these three, SOA, ZFC, or CoC? And which is the weakest? I means, the sentance "function f eventually outgrows all function provable recursive in SOA", "function f eventually outgrows all function provable recursive in ZFC", and "function f eventually outgrows all function provable recursive in CoC", which is strongest? {hyp/^,cos} (talk) 00:24, April 11, 2015 (UTC) :At the very least, ZFC is stronger than SOA for sure. I don't know much about the relationships between CoC/ZFC and CoC/SOA, and I think they might be open problems at the moment. CoC is quite obscure compared to the other two so I imagine it simply hasn't been explored much. -- ve 01:11, April 11, 2015 (UTC) ::I'm almost sure ZFC > CoC > SOA, but I don't know any references which would support this. LittlePeng9 (talk) 05:04, April 11, 2015 (UTC) :::Yeah, that's my guess too, but I don't think anyone's shown that ZFC > CoC or CoC > SOA. -- ve 07:36, April 11, 2015 (UTC) :::: No, I believe they have, even though I haven't located a reference yet. I believe it's been shown that the functions expressible is CoC are the functions that are provably recursive in higher order logic, so that would be a superset of the functions provably recursive in second order arithmetic and a subset of the functions provably recursive in ZFC. Certainly CoC is a stronger system than system F, and the functions expressible in system F are precisely the functions provably recursive in second order arithmetic. (See Second-order arithmetic for example) Deedlit11 (talk) 08:39, April 11, 2015 (UTC) :Here's statement that CoC is as strong as higher-order arithmetic, so CoC > SOA. {hyp/^,cos} (talk) 02:18, December 14, 2017 (UTC) Strength without some axioms Compared with ZFC, how strong (how weak) are these set theories, and how large are the PTOs of them? #ZFC without axiom of pairing #ZFC without axiom schema of specification #ZFC without axiom of power set #ZFC without axiom of union #ZFC without axiom schema of replacement #ZF #ZFC without axiom of infinity, plus "the empty set exists" Axiom of extensionality and axiom of regularity are "properties that sets should fit" and don't make new sets; they are not discussed here. For 1 and 2, they seem to be the same as ZFC because axiom of pairing and axiom schema of specification can be derived from other axioms. For 7, without axiom of infinity we can't get the existence of a set, so there is some modification. {hyp/^,cos} (talk) 01:27, September 22, 2018 (UTC)